the electron operator \(c_{i\sigma }\)  remains unchanged \(c_{i\sigma } \rightarrow c_{i\sigma }\) . This reveals an emergent local \(U(1)\)  gauge symmetry in the slave particle representation of electrons (at half filling the gauge symmetry is enlarged into \(SU(2)\)  [1]}, [2]}, [3]}, [4]}). The spinons and the holons interact with each other via the emergent \(U(1)\)  gauge field. Depending on the electron-electron interactions and the doping concentration, the \(U(1)\)  gauge field may get confined, resulting in a magnetically ordered phase or a superconducting phase[5]}. To begin with, we need to translate the interactions between the electrons and spins into the interactions between the slave particles.
